An Alternate to the Unified Distribution with Application to Breast Cancer Data

ABSTRACT

The Unified (θ) distribution appeared in [1], and in this short note we introduce a variant of this distribution. Further, a member of this family is shown to be significant in cancer modeling.

KEYWORDS

Bates distribution; Unified distribution; Cancer modeling

THE UNIFIED (θ) DISTRIBUTION

For x ∊(0,1) , the PDF is given by

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Remark 3.1. For details on the derivation of this family, see [1]

THE NEW FAMILY

Let X be Bates on (0, 1) [2], and denote its PDF as f x (x,n) and consider

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where k and ø are defined as in [1]. Suppose Q=q, and consider =a+(b-a)J, where the random variable J has density q then

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So, the CDF of Y is

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By differentiation the PDF of Y is

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So, if ø=1, then our density becomes

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The class of all distributions with the above density is denoted A Unified (a, b, θ)

Remark 4.1. The families A Unified (a, b, θ) and Unified (θ) (of Section 1) coincide if and only if a=0 and b=1

THE UNIFIED (θ) GENERATED FAMILY OF DISTRIBUTIONS

The A Unified(a, b, θ) generated family of distributions has CDF

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where 0 ≠θ ϵR, and K(x) is the CDF of some baseline distribution. Assuming the baseline distribution is Weibull with the following CDF

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for x, g, h>0, then from the integral we have the following

Proposition 5.1. The CDF of the A Unified (0, 1, θ) Weibull family is given by

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where x, g, h>0 and 0 ≠ ϵR
Obviously, the PDF can be obtained by differentiation. We write H AUW g h  (θ, , ), if H is a A Unified (a, b, θ) Weibull random variable. The family in the above Proposition is a good fit to real life data as shown below. The MLE estimates are obtained using the software MATHEMATICA (Figure 1) [3].

CONCLUDING REMARKS

In the present paper we introduced an alternate to the Unified distribution appearing in [1], and showed the Unified distribution is good in fitting real life data.

ACKNOWLEDGE

I am grateful to Oscar Alberto Quijano Acura of Concordia University, Montreal, Canada for pointing out an error in an earlier version of this paper.

REFERENCES

  1. Oscar AQA (2019) The unified distribution. Journal of Statistical Distributions and Applications 6: 13.
  2. Bates distribution. Wikipedia, The Free Encyclopedia.
  3. Jayakumar K, Girish BM (2017) T-Transmuted X family of distributions. STATISTICA 77(3): 251-276.

Article Type

Short Communication

Publication history

Received date: November 19, 2019
Published date: November 25, 2019

Address for correspondence

Clement Boateng Ampadu; 31 Carrolton Road, Boston, MA 02132-6303, USA

Copyright

©2019 Open Access Journal of Biomedical Science, All rights reserved. No part of this content may be reproduced or transmitted in any form or by any means as per the standard guidelines of fair use. Open Access Journal of Biomedical Science is licensed under a Creative Commons Attribution 4.0 International License

How to cite this article

Clement BA. An Alternate to the Unified Distribution with Application Breast Cancer Data. 2019 - 1(2) OAJBS.ID.000117.

Author Info

Clement Boateng Ampadu*

31 Carrolton Road, Boston, MA 02132-6303, USA

Figure 1: : The CDF of AUW (0.145698, 1.2849, 48.5111) fitted to the empirical distribution of the breast cancer data [3].

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